Seminars and Colloquia by Series

Boundary Value Problems for Nonlinear Dispersive Wave Equations

Series
PDE Seminar
Time
Tuesday, October 20, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Hongqiu Chen – University of Memphis
Under the classical small-amplitude, long wave-length assumptions in which the Stokes number is of order one, so featuring a balance between nonlinear and dispersive effects, the KdV-equation u_t+ u_x + uu_x + u_xxx = 0 (1) and the regularized long wave equation, or BBM-equation u_t + u_x + uu_x-u_xxt = 0 (2) are formal reductions of the full, two-dimensional Euler equations for free surface flow. This talk is concerned with the two-point boundary value problem for (1) and (2) wherein the wave motion is specified at both ends of a finite stretch of length L of the media of propagation. After ascertaining natural boundary specifications that constitute well posed problems, it is shown that the solution of the two-point boundary value problem, posed on the interval [0;L], say, converges as L converges to infinity, to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch [0;1) of the medium is disturbed at its finite end (the so-called wavemaker problem). In addition to its intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the quarter plane problem for both the KdV-equation and the BBM-equation.

Boundary layer associated with the Darcy-Brinkman-Boussinesq system

Series
PDE Seminar
Time
Tuesday, October 13, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Xiaoming Wang – Florida State University
We study the asymptotic behavior of the infinite Darcy-Prandtl number Darcy-Brinkman-Boussinesq model for convection in porous media at small Brinkman-Darcy number. This is a singular limit involving a boundary layer with thickness proportional to the square root of the Brinkman-Darcynumber . This is a joint work with Jim Kelliher and Roger Temam.

The Vlasov-Poisson System with Steady Spatial Asymptotics

Series
PDE Seminar
Time
Tuesday, September 29, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Stephen Pankavich – University of Texas, Arlington
We formulate a plasma model in which negative ions tend to a fixed, spatially-homogeneous background of positive charge. Instead of solutions with compact spatial support, we must consider those that tend to the background as x tends to infinity. As opposed to the traditional Vlasov-Poisson system, the total charge and energy are thus infinite, and energy conservation (which is an essential component of global existence for the traditional problem) cannot provide bounds for a priori estimates. Instead, a conserved quantity related to the energy is used to bound particle velocities and prove the existence of a unique, global-in-time, classical solution. The proof combines these energy estimates with a crucial argument which establishes spatial decay of the charge density and electric field.

Comparison principle for unbounded viscosity solutions of elliptic PDEs with superlinear terms in $Du$

Series
PDE Seminar
Time
Tuesday, September 22, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Shigeaki Koike – Saitama University, Japan
We discuss comparison principle for viscosity solutions of fully nonlinear elliptic PDEs in $\R^n$ which may have superlinear growth in $Du$ with variable coefficients. As an example, we keep the following PDE in mind:$$-\tr (A(x)D^2u)+\langle B(x)Du,Du\rangle +\l u=f(x)\quad \mbox{in }\R^n,$$where $A:\R^n\to S^n$ is nonnegative, $B:\R^n\to S^n$ positive, and $\l >0$. Here $S^n$ is the set of $n\ti n$ symmetric matrices. The comparison principle for viscosity solutions has been one of main issues in viscosity solution theory. However, we notice that we do not know if the comparison principle holds unless $B$ is a constant matrix. Moreover, it is not clear which kind of assumptions for viscosity solutions at $\infty$ is suitable. There seem two choices: (1) one sided boundedness ($i.e.$ bounded from below), (2) growth condition.In this talk, assuming (2), we obtain the comparison principle for viscosity solutions. This is a work in progress jointly with O. Ley.

Convergence properties of solutions to several classes of PDEs

Series
PDE Seminar
Time
Tuesday, September 15, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Zhang, Lei – University of Florida –
Many problems in Geometry, Physics and Biology are described by nonlinear partial differential equations of second order or four order. In this talk I shall mainly address the blow-up phenomenon in a class of fourth order equations from conformal geometry and some Liouville systems from Physics and Ecology. There are some challenging open problems related to these equations and I will report the recent progress on these problems in my joint works with Gilbert Weinstein and Chang-shou Lin.

On asymptotics, structure and stability for multicomponent reactive flows

Series
PDE Seminar
Time
Tuesday, September 8, 2009 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Konstantina Trivisa – University of Maryland, College Park
Multicomponent reactive flows arise in many practical applicationssuch as combustion, atmospheric modelling, astrophysics, chemicalreactions, mathematical biology etc. The objective of this work isto develop a rigorous mathematical theory based on the principles ofcontinuum mechanics. Results on existence, stability, asymptotics as wellas singular limits will be discussed.

Global Existence of a Free Boundary Problem with Non--Standard Sources

Series
PDE Seminar
Time
Tuesday, September 1, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Lincoln Chayes – UCLA
This seminar concerns the analysis of a PDE, invented by J.M. Lasry and P.L. Lions whose applications need not concern us. Notwithstanding, the focus of the application is the behavior of a free boundary in a diffusion equation which has dynamically evolving, non--standard sources. Global existence and uniqueness are established for this system. The work to be discussed represents a collaborative effort with Maria del Mar Gonzalez, Maria Pia Gualdani and Inwon Kim.

Analyticity in time and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow

Series
PDE Seminar
Time
Tuesday, August 25, 2009 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
David Hoff – Indiana University, Bloomington
We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. One important corollary is backwards uniqueness: if two such solutions agree at a given time, then they must agree at all previous times. Additionally, analyticity yields sharp estimates for the time derivatives of arbitrary order of solutions along particle trajectories. I'm going to integrate into the talk something like a "pretalk" in an attempt to motivate the more technical material and to make things accessible to a general analysis audience.

Nonlinear 4th order diffusion equations by optimal transport

Series
PDE Seminar
Time
Tuesday, May 5, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Giuseppe Savare – Università degli Studi di Pavia, Italy
Some interesting nonlinear fourth-order parabolic equations, including the "thin-film" equation with linear mobility and the quantum drift-diffusion equation, can be seen as gradient flows of first-order integral functionals in the Wasserstein space of probability measures. We will present some general tools of the metric-variational approach to gradient flows which are useful to study this kind of equations and their asymptotic behavior. (Joint works in collaboration with U.Gianazza, R.J. McCann, D. Matthes, G. Toscani)

Steady Water Waves with Vorticity

Series
PDE Seminar
Time
Tuesday, April 14, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Joy Ko – Brown University, Providence
I will talk about the highlights of a collaborative and multidisciplinary program investigating qualitative features of steady water waves with vorticity in two dimensions. Computational and analytical results together with data from the oceanographic community have resulted in strong evidence that key qualitative features such as amplitude, depth, streamline shape and pressure profile can be fundamentally affected by the presence of vorticity. Systematic studies of constant vorticity and shear vorticity functions will be presented.

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